CP Violation in Higher Dimensional Gauge Theories

@ A series of seminars at RIKEN

(Dec. 5, ’09)

C.S. Lim （林　青司） 　　　 (Kobe University)

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I. Introduction 　 In spite of the great success of Kobayashi-Maskawa model, the origin of CP violation still seems to be not conclusive .

In D = 4 space-time, the mechanism of CP violation: ・ explicit (hard) breaking due to complex Yukawa coupling a la K-M ・ spontaneous (soft) breaking due to complex VEV’s of Higgs (T.D. Lee, S. Weinberg)

We may have some new mechanism to break CP, once space-time is extended, as in superstring theory or Kaluza-Klein type theories. In fact, CP violation due to compactification of extra space was discussed: C.S. Lim, Phys. Lett. B256(’91)233 (A.Strominger and E. Witten, Commun. Math Phys. 101(’85)231); T. Kobayashi and C.S.L., P.hys. Lett. B343(’95)122 (in orbifold string theory).

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However, e.g. in string theory or 10 dim. SUSY Y.-M. theory, as its low energy limit, to break CP is a non-trivial issue, as the theory does not have complex parameter, to start with.

(N.B.) ・ The scenario of Gauge-Higgs Unification (GHU) , as an attractive candidate of New Physics, also has the same problem.10 dim. SUSY Y.-M. theory is a sort of GHU, where Higgs originates form the extra space component of gauge field. 　・ In GHU, the VEV of the Higgs, Wilson line phase (Hosotani mechanism), can be a new source of the CP violation.

On the other hand, if it succeeds, it may provide a new type of mechanism of CP violation.

The condition for CP violation may be utilized in order to select viable models and/or the manner of compactification.

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(Higher dimensional C, P transf.s should be modified : C.S. L. ’91 )

We can easily find C matrix, for instance, in higher dim. space-time, so that it satisfies . ・ Such defined higher dimensional C, P transf.s , however, do not correspond to the 4-dimensional ones, in general, and some modification is necessary. ・ Interestingly, the modified CP transformation act on the extra space coordinates non-trivially: it acts as a complex conjugation of the complex homogeneous coordinates for the extra space. ・ If the compactfied space has “complex structure”, the breaking of CP can be realized.

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Take D=6 case for the illustrative purpose. In the basis, where 6D spinor decomposes into two 4-D spinors,

The C matrix is found not to reduce to ordinary 4-D transf.,

Modifying C and P as

gamma matrices are given as

, because of .

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Accordingly the transformation properties of a vector

is uniquely determined and we find:

(N.B.) The reason of the peculiar transf. under C was that extra dimensional gamma matrices are half symmetric and half anti-symmetric.

Thus introducing, a complex coordinate as

CP transf. is nothing but a complex conjugation:

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This peculiar property persists for higher (even) dimensions, as is easily seen by iterative construction:

For instance, in 10D

(N.B.) In this case, C should not be modified, as Majorana fermion exists in D = 2 ,4 (mod 8), and P causes the complex conjugation.

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Consider Type-I superstring theory with 6-dimensional Calabi-Yau manifold defined by a quintic polynomial for the coordinates of CP4 ,

CP is broken only when the coefficient C is complex, since otherwise the above defining equation is invariant under

・ In fact, resultant Yukawa couplings is known to have a CP violating phase for complex C (M. Matsuda, T. Matsuoka, H. Mino, D. Suematsu and Y. Yamada, Prog. Theor. Phys. 79(’88)174). 　・ If we impose phenomenological requirement, of no FCNC at the tree level, the CP phase disappears, unfortunately.

“4 generation model”

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Our purpose: To realize CP violation in the framework of higher dimensional field (gauge) theories, not string theories, with much simpler compact spaces, such as a circle or orbifold. More precisely, what we have in our mind as the higher dimensional gauge theory is GHU scenario. Note that as long as higher dimensional gauge theory itself is CP invariant, without phase, CP violation should be a sort of “spontaneous” breaking.

We have discussed two possibilities: 1. CP violation due to compactification (Z4 orbifold) 　　 (w./ N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-

ph] ) 2. CP violation due to the VEV of the Higgs (w./ Y. Adachi and N. Maru, arXiv:0905.1022 [hep-ph] , Phys. Rev. D 80(‘09)055025)

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I. Gauge-Higgs unification (GHU)

unification of gravity (s=2) & elemag (s=1) (A. Einstein)

Kaluza-Klein theory

unified theory of gauge (s=1) & Higgs (s=0) interactions

“Gauge-Higgs unification”

: realized in higher dimensional gauge theory

4D space-time4D gauge-field Higgs

5D gauge field

extra dimension

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the idea of gauge-Higgs unification itself is not new:

・ N.S. Manton, Nucl. Phys. 58(’79)141.

・ Y. 　 Hosotani, Phys. Lett. B126 (‘83) 309 : ``Hosotani mechanism”

The scenario was revived:

・ H. Hatanaka , T. Inami and C.S.L., Mod. Phys. Lett. A13(’98)2601( the main points )

・ The quantum correction to mH is finite because of the higher dimensional gauge symmetry → A new avenue to solve the hierarchy problem without invoking SUSY

・ The sum over all K-K modes is essential to get the finite （ for arbitrary dimensions ） Higgs mass

(N.B.) The scenario may also shed some light on the arbitrariness problem in the interactions of Higgs.

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・ Little Higgs model : 4D theory, where G/H of global symmetry provides Higgs as a N-G, may be “dual” to 5D GHU, where Ay

associated with G/H of higher dimensional local gauge symmetry provides Higgs (holographic principle).

II. Issues related to GHU

・ dimensional deconstruction (N. Arkani-Hamed, A.-G. Cohen, H. Georgi, Phys.Lett. B513(’01)232) : latticized 5D gauge theory , @ N → ∞ limit, the effective potential for H coincides with what we obtained.

・ GHU may be related even with the supersymmetry (!)

We have demonstrated that N=2 quantum mechanical (QM) SUSY is hidden in the gauge-Higgs unification scenario, which is due to Higgs-like mechanism for massive K-K modes. (On R-S background, the ``warp-factor” just corresponds to a superpotential .)

(T. Nagasawa, M. Sakamoto, H. Sonoda and C.S.L., Phys. Rev. D72(‘05) 064006)

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・ (ultra) natural inflation (N. Arkani-Hamed, H.-C. Cheng, P. Creminelli and L. Randall, Phys.Rev.Lett. 90(’03)221302; T. Inami, Y. Koyama, S. Minakami &C.S.L., Progr. Theor. Phys. (09), to appear) : Ay

(0) may be a natural candidate for the inflaton, as the local gauge symmetry stabilizes the potential under the quantum correction

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・ “Minimal GHU standard model”: SU(3) on M4 x (S1/Z2) (Kubo, C.S.L. and H. Yamashita, Mod. Phys. Lett. A17(’02)2249)

SU(3) → SU(2) x U(1) breaking due to non-trivial Z2-parity assignment (Kawamura):

Zero-modes of Gauge-Higgs sector :

(N.B.) In the GHU, gauge group should be enlarged, as the Higgs belongs to adjoint repr., while SM Higgs is SU(2) doublet.Recall that in the heterotic string theory, Higgs belonging to the fundamental repr. of E6 comes from adjoint repr. of E8 .

Exactly what we need for the SU(2) x U(1) SM !

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IV. CP violation due to compactification 　　 (w./ N. Maru and K. Nishiwaki, arXiv:0910.2314 [hep-

ph] ) How to break CP symmetry is a challenging issue in the scenario of GHU, where the Higgs interactions are governed by gauge principle.

One of a few possibilities to break CP symmetry is to invoke to the manner of compactificaion, which determines the vacuum state of the theory: “spontaneous breaking”.

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In Type I superstring theory, CP is broken only when the coefficient C to define Calabi-Yau manifold, complex, since otherwise the above defining equation is invariant under

In our paper we consider much simpler compactification; we discuss the CP violation in the 6-dimensional U(1) GHU model due to the compactification on the orbifold

We easily know that CP transfomration is not compatible with the condition of orbifolding.

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In terms of a complex coordinate

the orbifold condition is written as

After the CP transf.,

the condition reads as

Thus, CP tranf. is not compatible with orbifolding condition , and CP symmetry is broken.

: “orientation-changing operator” (Strominger and Witten)

(orbifolding) (CP trensf.)

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The model: 6D QED

Z4 orbifold conditions:

(double ) K-K mode expansions:

The zero-mode sector recovers ordinary QED, with

The presence of factor i signals the CP violation.

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The mass eigenstates for fermions:

where ,

By use of a freedom of unitary transformation due to the mass degenaracy between 　　　　　 , the roles of L and R can be changes. ← parity symmetry

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The interaction vertices of non-zero K-K photons:

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Even the interaction vertices for non-zero K-K photons generally have CP violating phases:

The EDM of electron, as a typical CP violating observable, however, is found to vanish at 1-loop level. Unfortunately, we anticipate that we cannot get a non-vanishing contributions even at higher loops.

・ Such obtained CP violating phases are confirmed to survive even after the re-phasing of the fields. ・ we also have identified “Jarlskog parameter” in our model.

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Or, if we recall that

P symmetry is not violated by the compactification, as is naively expected in QED.

Since, EDM necessitates both of P and CP violations, we anticipate EDM vanishes in our model, although we expect EDM will get contributions in a realistic theory including the SM, since P should be violated anyway in such a realistic theory.

This comes from the L-R symmetry due to the freedom of unitary transformation.

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2. CP violation due to the VEV of the Higgs (w./ Y. Adachi and N. Maru, arXiv:0905.1022 [hep-ph] , Phys. Rev. D 80(‘09)055025)

Another possibility to break CP is due to the VEV of some field which has odd CP eigenvalue. We argue that the VEV of the Higgs , or the VEV of Wilson-loop plays the role ( : “timeon” ?). We show that neutron EDM gets contribution already at 1-loop level in the model, though we assume the presence of only 1 generation.(The model) 5-D SU(3) GHU model compactified on an orbifold with a massive bulk fermion in a fundamental representation.

In this case, the orbifold is too simple to break CP, thus only possibility seems to be due to

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(N.B.) To get EDM, both P and CP have to be broken. P symmetry, however, is broken anyway by the orbifolding.

In 5D CP transf. can be defined just as in the 4D case:

The CP transf. is known to be consistent with the orbifolding condition as and commute with each another:

Correspondingly, the transformations of space-time and fields are fixed as,

Thus we realize that has odd CP eigenvalue and the VEV may lead to CP violation.

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Actually, when the Z2 – odd bulk mass term 　　 is switched off , we can perform a chiral rotation for , so that the coupling of becomes scalar type and therefore has even CP eigenvalue. Hence, to get physical CP violating effects, the interplay between the VEV and the bulk mass is crucial. (The neutron EDM) In this mechanism of CP violation, EDM appears already at 1-loop level, though we have only 1 generation.

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・ We have confirmed that EDM appears only when both of M and the VEV of Ay are non-zero.

・ Let us note that in K-M model, the EDM arises only at the 3-loop order.

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with the experimental upper bound on the EDM, we get the lower bound on the compactification scale,

Comparing the contribution of the non-zero KK modes